In: Statistics and Probability
"Durable press" cotton fabrics are treated to improve their recovery from wrinkles after washing. Unfortunately, the treatment also reduces the strength of the fabric. The breaking strength of untreated fabric is normally distributed with mean 51.8 pounds and standard deviation 2.6 pounds. The same type of fabric after treatment has normally distributed breaking strength with mean 19.1 pounds and standard deviation 2 pounds. A clothing manufacturer tests 3 specimens of each fabric. All 6 strength measurements are independent. (Round your answers to four decimal places.)
Since the specimens are independent, the sample mean of the
breaking strengths of the 5 untreated specimens is normally
distributed with mean 51.9 pounds and standard deviation
2.9/sqrt(5) pounds.
So the probability that the sample mean of the breaking strengths
of the 5 untreated specimens exceeds 50 pounds is
P(Z > (50 - 51.9)/(2.9/sqrt(5))) = P(Z > -1.47) = P(Z <
1.47) = 0.9292.
b) Since the specimens are independent, the sample mean of the
breaking strengths of the 5 untreated specimens is normally
distributed with mean 51.9 pounds and standard deviation
2.9/sqrt(5); the sample mean of the breaking strengths of the 5
treated specimens is normally distributed with mean 20.5 pounds and
standard deviation 1.9/sqrt(5) pounds.
Because the untreated and treated sample means are independent as
well, the difference of the untreated minus treated sample means is
normally distributed with mean 51.9 - 20.5 = 31.4 pounds and
standard deviation sqrt[(2.9/sqrt(5))^2 + (1.9/sqrt(5))^2] =
sqrt(2.404) pounds (recall that the variance of a sum or difference
of *independent* random variables is the *sum* of their variances,
so it follows that the standard deviation is the square root of the
sum of the squares of their standard deviations).
So the probability that the sample mean of the breaking strengths
of the 5 untreated specimens is at least 25 pounds greater than the
sample mean of the breaking strengths of the 5 treated specimens
is
P(difference of the untreated minus treated sample means >=
25)
= P(Z >= (25 - 31.4)/sqrt(2.404))
= P(Z >= -4.13)
= P(Z <= 4.13), which is very nearly 1.